FEDERAL UNIVERSITYOF ABC

DOCTORAL THESIS

ASPECTS OF BLACK HOLE BEYOND GENERAL RELATIVITY: EXTRA DIMENSIONS, HORIZON WAVE FUNCTION AND APPLICATIONS

Author: Supervisor: Rogério Teixeira Cavalcanti Dr. Roldão da Rocha Jr.

Thesis submitted to the Federal University of ABC (UFABC) in fulfillment of the requirements for the degree of in the field of Theoretical Physics. arXiv:1902.10128v1 [gr-qc] 25 Feb 2019

Santo André Brazil 2017 Sistema de Bibliotecas da Universidade Federal do ABC Elaborada pelo Sistema de Geração de Ficha Catalográfica da UFABC com os dados fornecidos pelo(a) autor(a).

Cavalcanti, Rogério Teixeira Aspects of Black Hole Physics Beyond General Relativity : extra dimensions, horizon wave function and applications / Rogério Teixeira Cavalcanti. — 2017.

123 fls. : il.

Orientador: Roldão da Rocha Jr.

Tese (Doutorado) — Universidade Federal do ABC, Programa de Pós Graduação em Física, Santo André, 2017.

1. Buracos Negros. 2. Dimensões Extras. I. Rocha Jr., Roldão da. II. Programa de Pós-Graduação em Física, 2017. III. Título. iii Acknowledgements

There are many people directly and indirectly related to this work who deserve more than a few words preceding the main text. My wife is surely the person who suffered more due to my constant absence and lack of time. I am very grateful for all the support she gave me, always helping me to become a better person. Thank you and I love you Juliana! My parents also supported me all this time and I’m very grateful for that. Thank you mom and dad (in memoriam). I had the fortune of meeting some special people during my time as a graduate student. With them I had many discussions about physics, mathematics, science and life in general. I do not intend to mentioning everybody who deserves be- cause I would surely miss a few of them, but José, Allan, André, Kelvyn, Armando, Bruno, Marina, Luis and Daniel are some of these who have helped me to better understand the meaning of friendship and science. Over the year of 2015 I have been collaborating with professor Roberto Casadio, who hosted me at the Bologna university. Roberto was a great host and I have learnt a lot with him. Thank you! Finally my advisor, professor Roldão da Rocha. I have learnt much more than how I should carry on a research and technical stuff with him. I am sure his other students have the same feeling. Sometimes he treats us as students, some times as friends, but always being very professional, kind and ready for helping. For me it is hard imagining a better advisor. Thank you very much professor Roldão! iv

FEDERAL UNIVERSITY OF ABC Center of Natural and Human Sciences Abstract

This work is devoted to investigate some consequences of black holes physics beyond the domain of general relativity, mainly in effective extra dimensional mod- els. The investigation is carried along three gravitational effects, namely the Hawk- ing radiation, the strong deflection of gravitational lensing and the formation of quantum black holes. A cosmological thick brane solution is also investigated. Ef- fective theories and models provide a prominent approach for testing the limits of known theories and show what would be expected beyond that. Based on such idea we have used effective models for finding deviations of general relativity as- sociated to each of the mentioned phenomena. Keywords: Black Holes, Extra Dimensions, Gravitational Lensing, Quantum Black Holes, Generalized Uncertainty Principle. v

UNIVERSIDADE FEDERAL DO ABC Centro de Ciências Naturais e Humanas Resumo

Neste trabalho foram investigadas algumas conseguências da física de buracos negros em teorias cujo domínio está além do domínio da relatividade geral, em especial em teorias efetivos com dimensões extras. A investigação foi em substan- cialmente conduzida baseando-se em três efeitos gravitacionais, a saber, a radiação Hawking, o regime de deflexão forte de lentes gravitacionais e a formação de bura- cos negros quânticos. Uma solução de modelo cosmológico imerso em uma brana espessa foi também investigada. Modelos e teorias efetivas fornecem meios para testar os limites de validade de teorias conhecidas e indicam o que deveríamos esperar além desses limites. Baseado nessa ideia foram usados alguns modelos efe- tivos para estudar efeitos não previstos pela relatividade geral, associados a cada um dos fenômenos mencionados. Palavras Chave: Buracos Negros, Dimensões Extras, Lentes Gravitacionais, Bu- racos Negros Quânticos, Princípio da Incerteza Generalizado. vi

Contents

List of Figures viii

List of Tables x

1 Introduction 2 1.1 Results and Thesis Organization ...... 3

2 Preliminaries 5 2.1 A Glimpse of Black Holes in General Relativity ...... 6 2.1.1 Asymptotically Flat Solutions ...... 7 2.1.2 Horizons ...... 10 2.1.3 General Properties ...... 12 2.2 Black Hole Mechanics and Thermodynamics ...... 14 2.2.1 Hawking Radiation ...... 16 2.3 Beyond General Relativity ...... 18 2.3.1 Why Black Holes? ...... 20

3 Elko Dark Spinors Emission Through Hawking Radiation 21 3.1 The Hamilton-Jacobi Method ...... 22 3.1.1 Bosonic Emission ...... 22 3.1.2 Fermionic Emission ...... 25 3.2 The Kerr-Sen Dilaton-Axion Black Hole ...... 30 3.3 Elko Dark Spinors ...... 31 3.4 Tunneling From Kerr-Sen Dilaton-Axion ...... 32 3.4.1 Dirac Spinors ...... 32 3.4.2 Elko Dark Spinors ...... 35

4 Extra Dimensions and Braneworld FRW 39 4.1 Overview ...... 39 4.1.1 Braneworld ...... 40 4.1.2 Braneworld Cosmology ...... 41 4.2 Kaluza-Klein Modes ...... 41 4.3 Thin Braneworld Models ...... 43 4.3.1 Compact Flat Extra Dimensions ...... 44 4.3.2 Warped Extra Dimensions ...... 45 4.3.3 Effective Field Equations ...... 46 4.4 Thick Branes ...... 48 4.4.1 Topological Defects and Domain Wall Branes ...... 49 4.4.2 Thick Brane FRW Solution ...... 51 4.5 Braneworld Black Holes ...... 60 4.5.1 The Schwarzchild Extra Dimensional Solution ...... 61 4.5.2 The Casadio-Fabbri-Mazzacurati (CFM) Solutions ...... 62 4.5.3 The Minimal Geometric Deformation (MGD) Solution . . . . 63 vii

5 Strong Deflection Limit in Gravitational Lensing 66 5.1 Exact Solutions, the Weak and the Strong Deflection Limit ...... 68 5.2 The SDL coefficients for Spherically Symmetric Space-Times . . . . . 73 5.2.1 SDL Coefficients for α(θ) ...... 75 5.3 Deflection and Coefficients of the MGD and CFM Solutions . . . . . 76 5.3.1 MGD ...... 77 5.3.2 CFM I & II ...... 78 5.4 Observables in the strong deflection limit ...... 80

6 Quantum Black Holes and the Horizon Wave Functions Formalism 85 6.1 Minimum Length and GUP ...... 85 6.2 The Horizon Wave Functions Formalism ...... 87 6.2.1 Gaussian Sources ...... 89 6.2.2 GUP Revisited ...... 92 6.3 HWF and Extra Dimensions ...... 95 6.3.1 Black Hole Probability ...... 97 6.3.2 GUP From Higher Dimensional HWF ...... 98

7 Conclusion 103

A List of Published Papers 105

B Explicit Calculations of the Section 4.4.2 106

Bibliography 116 * viii

List of Figures

2.1 Depicture of a spherically symmetric star collapsing into an ABH. . .7 2.2 Depicture of Kerr or Kerr-Newman black hole and its ergosphere. . . 10 2.3 Representation of the displacement p 7→ p0 = p+εV producing δA 7→ δA0. In this plot S appears as a closed line, whereas it is actually 2- dimensional surface. Figure taken from [1]...... 11 2.4 Intersection of the null cone of p ∈ S and the orthogonal tangent ⊥ space [Tp(S) ] defining the directions k and l. Figure taken from [1]. 11 2.5 Propagation of null wafefronts emitted on the tick circle, inside a trapping surface. Both, ingoing and outgoing wavefronts, converge to the surface interior ...... 12 2.6 Possible asymptotically flat black holes in GR...... 13 2.7 (M,R) diagram showing different physics domains. In logo scale, the mass correspondig to -5 (10−5 g) is the Planck mass and length equals to -33 (10−33 cm) i